Irregular packing: MILP model based on a polygonal enclosure

Santoro, Miguel Cezar; Lemos, Felipe Kesrouani

doi:10.1007/s10479-015-1971-9

Cited by 13 publications

(15 citation statements)

References 36 publications

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“…The biggest reduction of nodes explored to prove optimality was obtained for instance santoro1, where NFP-CM explored 185.36 times the amount of nodes of improved NFP-CM. Furthermore, improved NFP-CM solved all instances santoro faster than the original paper [11]. We also highlight improved NFP-CM is the first MIP model to prove optimality for instances threep3, threep3w9 and fu in less than 1,200 seconds.…”

Section: Resultsmentioning

confidence: 83%

“…Alvarez-Valdes et al [1] defined some structures used in [5] and developed a branch-and-bound algorithm to solve the model. Santoro and Lemos [11] proposed a MIP model to convex polygons with some limitations: the polygons can have up to eight edges and their opposite edges must be parallel. Cherri et al [3] proposed two models that allow rotation of pieces.…”

Section: Introductionmentioning

confidence: 99%

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MIP models for the irregular strip packing problem: new symmetry breaking constraints

2017

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Abstract. The irregular strip packing problem consists in minimizing the length used to cut a set of pieces from a board with fixed width. Recently, a mixed integer programming model was proposed for the problem, but it may allow a large number of symmetric solutions. In this paper, new symmetry breaking constraints are proposed to improve the model. Computational experiments were performed for instances with convex pieces. The results show the proposed formulation is better than the previous one for most instances, since it improves lower bounds and reduces run-time and number of nodes explored to prove optimality.

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Section: Resultsmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

MIP models for the irregular strip packing problem: new symmetry breaking constraints

2017

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“…The biggest reduction of nodes explored to prove optimality was obtained for instance santoro1, where NFP-CM explored 185.36 times the amount of nodes of iNFP-CM. Furthermore, despite the proposed model is more general, it solved all santoro instances faster than the original paper Santoro and Lemos (2015).…”

Section: Resultsmentioning

confidence: 97%

“…In addition, the authors propose a mixed integer programming model derived from a compaction model proposed by Gomes and Oliveira (2006). Santoro and Lemos (2015) proposed a MIP model to convex polygons with some limitations: the polygons can have up to eight edges and their opposite edges must be parallel. In we proposed two models that allow rotation of pieces.…”

Section: Introductionmentioning

confidence: 99%

Mathematical models and heuristic methods for nesting problems

Mundim¹

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Irregular cutting and packing problems, with convex and non-convex polygons, are found in many industries such as metal mechanics, textiles, of shoe making, the furniture making and others. In this thesis we study the two-dimensional version of these problems, where we want to allocate a set of items, without overlap, inside one or more containers, limited or unlimited, so as to optimize an objective function. In this document we study the knapsack problem, placement problem, strip packing problem, cutting stock problem and bin packing problem. For these problems, the heuristic methods and mathematical programming models are proposed and presented very promising results, surpassing in many cases the best results in the specialized literature. This thesis is organized as follows. In Chapter 1, we present a review of the studied problems, the value proposition for this thesis with the main contributions and ideas. In Chapter 2, we propose a metaheursitic for the strip packing problem with irregular items and circles. Then, in Chapter 3, we present a generic heuristic for the allocation of irregular items that may be weakly or strongly heterogeneous and will be allocated in a container (output maximization problems) or multiple containers (input minimization problems). In Chapter 4, we propose a solution method for the cutting stock problem with deterministic demand and stochastic demand. In Chapters 5 and 6, we present mathematical programming models for the strip packing problem. Finally, in Chapter 7, we present a conclusion and a concise direction for future works.

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“…The most recent models to irregular strip packing problem are Cherri et al (2016), Leao et al (2016), Santoro and Lemos (2015). To combine the accuracy of continuous models with the advantages of the dotted-board model, Leao et al (2016) proposed a semi-continuous model where the items can be placed in continuous positions in the x-axis and in discrete positions in the y-axis.…”

Section: List Of Tablesmentioning

confidence: 99%

Irregular and quasi-polyomino strip packing problems

Rodrigues¹

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The irregular strip packing problem consists in the cutting of a set of two-dimensional pieces from an object of fixed width using the minimum possible length. Despite its economic importance for many industries, because of its resolution difficulty few exact methods have addressed this problem. Recently, a mixed integer programming model in which pieces are placed on a grid has been proposed. Although the model has proved the optimality for some large instances, it has a large number of non-overlap constraints, which grows quickly according to the discretization resolution and number of distinct pieces. This thesis proposes a clique covering model to reduce the number of constraints and improve the linear relaxation. The coverings were obtained by a heuristic developed by the author. The model has outperformed the previous model in most evaluated instances and obtained an optimal solution for instances with up to 25 pieces (22 distinct pieces) subject to grid discretization. Recently, another mixed integer programming model was proposed for the problem, but it may allow a large number of symmetric solutions. In this thesis, new symmetry breaking constraints are proposed to improve the model. Computational experiments were performed for instances with convex pieces. The results show the proposed formulation is better than the previous one for most instances, since it improves lower bounds and reduces run-time and number of nodes explored to prove optimality. A particular case of an irregular item is a polyomino. A polyomino is a set of unit squares connected by joining one of their edges. A quasi-polyomino is a polyomino generalization, since it is a subset of not necessarily connected squares obtained from an equidistant raster grid. Quasi-polyomino cutting and packing problems have many real applications, e.g., leather cutting, sheet metal stamping, design of printed circuit boards and layout of magazines and newspapers. In this thesis, we study the quasi-polyomino strip packing problem. We propose two integer programming models for the problem and evaluate them using state-of-the-art solvers. We evaluate the models using instances taken from the literature and both models obtained good results, solving to optimality an instance with 320 items (20 distinct items) on a strip of dimensions 44x50. As expected, we found more optimal solutions when there are no rotation and flips and when the dimensions of the items are small.

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Irregular packing: MILP model based on a polygonal enclosure

Cited by 13 publications

References 36 publications

MIP models for the irregular strip packing problem: new symmetry breaking constraints

MIP models for the irregular strip packing problem: new symmetry breaking constraints

Mathematical models and heuristic methods for nesting problems

Irregular and quasi-polyomino strip packing problems

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